Problem 48 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A certain mass of an ideal gas at \(9 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) is first heated to \(131^{\circ} \mathrm{C}\) at constant volume and then the amount of the gas is increased by \(50 \%\) at constant volume and temperature. The final pressure of the gas becomes (a) 9 atm (b) \(4.5 \mathrm{~atm}\) (c) 18 atm (d) \(13.5 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The final pressure of the gas is 13.5 atm.

Step 1: Use the Ideal Gas Law for the initial state

For the initial state at 9 atm and 30°C (which is 303.15 K after converting from Celsius to Kelvin), the ideal gas law can be expressed as: \( P_1 V_1 = n_1 R T_1 \).

Step 2: Use the Ideal Gas Law for the final state after heating

After heating the gas to 131°C (which is 404.15 K), at constant volume, the ideal gas law gives us: \( P_2 V_1 = n_1 R T_2 \), since the volume and the amount of gas remain the same.

Step 3: Relate the initial and final pressures using Charles's Law

Since the volume and amount of gas remain constant during the heating process, Charles's Law (\( P_1/T_1 = P_2/T_2 \)) allows us to relate the initial and final pressures as \( P_2 = P_1 * (T_2 / T_1) \). Substituting the known values, we calculate the final pressure after heating.

Step 4: Calculate the effect of increasing the amount of gas by 50%

After heating, when the amount of gas is increased by 50% at constant volume and temperature, the final pressure \(P_3\) can be found using the direct proportion between the amount of gas and pressure \( (P_2/n_1 = P_3/n_2) \). Since \( n_2 = 1.5 \times n_1 \), we can find the new pressure.

Step 5: Calculate the final pressure

Calculate \(P_3\) using the relation \( P_3 = 1.5 \times P_2 \) from Step 4. Finalize your solution by providing the correct answer from the given options.

Key Concepts

Charles's Law

Charles's Law is a foundational principle in gas law studies, describing how gases tend to expand when heated. In simpler terms, it holds that the volume of an ideal gas is directly proportional to its temperature, provided the pressure and the amount of gas remain constant. This can be mathematically expressed using the relation \[\begin{equation}\frac{V_1}{T_1} = \frac{V_2}{T_2},\end{equation}\]where V represents volume, T indicates temperature in Kelvin (K), and subscripts 1 and 2 refer to the initial and final states of the gas, respectively. To use Charles's Law in practice, as in our textbook example, we can rearrange it to find the new pressure (\[\begin{equation}\frac{P_1}{T_1} = \frac{P_2}{T_2},\end{equation}\]), assuming the volume remains constant. By plugging in the known temperatures and initial pressure, we can solve for the final pressure after heating.

Gas Laws in Chemistry

Gas Laws in Chemistry provide the framework for understanding the behavior of gases under various conditions. The most commonly referred to gas laws include Boyle's Law, Charles’s Law, Gay-Lussac's Law, Avogadro's Law, and the combined Gas Law. Finally, these laws are all encompassed by the Ideal Gas Law, which is expressed by the equation \[\begin{equation}PV = nRT.\end{equation}\]In this equation, P stands for pressure, V is for volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. When dealing with problem-solving in chemistry, understanding these laws helps predict how a gas will react when subjected to changes in temperature (\[\begin{equation}T,\end{equation}\]), volume (\[\begin{equation}V,\end{equation}\]), or pressure (\[\begin{equation}P.\end{equation}\]). For example, in the given problem, understanding Charles's Law allowed us to determine the effect of temperature on pressure while keeping the volume constant.

Problem Solving in Physical Chemistry

Problem Solving in Physical Chemistry typically involves a logical step-by-step approach to apply chemical principles, like gas laws, to find a solution. It starts with identifying the known variables and the chemical principle that applies to the situation. For instance, in our textbook exercise, the steps involved understanding the Ideal Gas Law and Charles’s Law, then using these relationships to calculate changes in pressure due to temperature changes and changes in the amount of gas. Finally, numerical substitution and algebraic manipulation lead to the answer.An effective problem-solving strategy includes:

• Reading the problem carefully and noting what is being asked.
• Writing down all the given information and organizing data.
• Selecting the appropriate formula(s) related to the concept.
• Simplifying the problem by canceling out constants or combining equations.
• Performing the calculations accurately.
• Checking the result against the original problem to make sure it is reasonable.

Applying this methodology can enhance comprehension and retention, ultimately helping students tackle even complex physical chemistry challenges with confidence.